and select the \(\delta \) -SMI posterior \(p^{(k)}_{\delta ^*}\)  for analysis.
In general the ELPD must be estimated as \(p^*\)  is unknown.
In Section REF  (a simple synthetic example) we calculate the ELPD exactly. In Section REF  we use LOOCV to estimate the ELPD (for the \(Z\) -data alone). In Section REF  we use the WAIC to estimate the ELPD for the \(Y\)  and \(Z\)  data separately using the methods of [1]}.
